Calculus Examples

$y = 5 x^{2} e^{3 x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (5 x^{2} e^{3 x})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2} e^{3 x}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2} e^{3 x}]$ .

$5 \frac{d}{d x} [x^{2} e^{3 x}]$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = e^{3 x}$ .

$5 (x^{2} \frac{d}{d x} [e^{3 x}] + e^{3 x} \frac{d}{d x} [x^{2}])$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 3 x$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $3 x$ .

$5 (x^{2} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x]) + e^{3 x} \frac{d}{d x} [x^{2}])$

Step 3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$5 (x^{2} (e^{u} \frac{d}{d x} [3 x]) + e^{3 x} \frac{d}{d x} [x^{2}])$

Step 3.3.3

Replace all occurrences of $u$ with $3 x$ .

$5 (x^{2} (e^{3 x} \frac{d}{d x} [3 x]) + e^{3 x} \frac{d}{d x} [x^{2}])$

Step 3.4

Differentiate.

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Step 3.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$5 (x^{2} (e^{3 x} (3 \frac{d}{d x} [x])) + e^{3 x} \frac{d}{d x} [x^{2}])$

Step 3.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$5 (x^{2} (e^{3 x} (3 \cdot 1)) + e^{3 x} \frac{d}{d x} [x^{2}])$

Step 3.4.3

Simplify the expression.

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Step 3.4.3.1

Multiply $3$ by $1$ .

$5 (x^{2} (e^{3 x} \cdot 3) + e^{3 x} \frac{d}{d x} [x^{2}])$

Step 3.4.3.2

Move $3$ to the left of $e^{3 x}$ .

$5 (x^{2} (3 \cdot e^{3 x}) + e^{3 x} \frac{d}{d x} [x^{2}])$

Step 3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$5 (x^{2} (3 e^{3 x}) + e^{3 x} (2 x))$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$5 (x^{2} (3 e^{3 x})) + 5 (e^{3 x} (2 x))$

Step 3.5.2

Combine terms.

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Step 3.5.2.1

Multiply $3$ by $5$ .

$15 (x^{2} (e^{3 x})) + 5 (e^{3 x} (2 x))$

Step 3.5.2.2

Multiply $2$ by $5$ .

$15 x^{2} e^{3 x} + 10 e^{3 x} x$

Step 3.5.3

Reorder terms.

$15 e^{3 x} x^{2} + 10 e^{3 x} x$

Step 3.5.4

Reorder factors in $15 e^{3 x} x^{2} + 10 e^{3 x} x$ .

$15 x^{2} e^{3 x} + 10 x e^{3 x}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 15 x^{2} e^{3 x} + 10 x e^{3 x}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 15 x^{2} e^{3 x} + 10 x e^{3 x}$